The free tracial post-Lie-Rinehart algebra of planar aromatic trees for the design of divergence-free Lie-group methods

Aromatic Butcher series were successfully introduced for the study and design of numerical integrators that preserve volume while solving differential equations in Euclidean spaces. They are naturally associated to pre-Lie-Rinehart algebras and pre-Hopf algebroids structures, and aromatic trees were shown to form the free tracial pre-Lie-Rinehart algebra. In this paper, we present the generalisation of aromatic trees for the study of divergence-free integrators on manifolds. We introduce planar aromatic trees, prove that they span the free tracial post-Lie-Rinehart algebra, and apply them for deriving new Lie-group methods that preserve geometric divergence-free features up to a high order of accuracy.